A vessel of depth 2h is half filled with a liquid of refractive index 2\sqrt 2 and the upper half with another liquid of refractive index \sqrt 2 . The liquids are immiscible. The apparent depth of the inner surface of the bottom of vessel will be :

Geometrical Optics PYQ — JEE Main 2020 · 9 Jan · Shift 1 · Q66 with Solution

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Correct answer: C

The vessel has total depth $2h$ and is filled in two layers:

We need the apparent depth of the bottom surface as seen from air.

For normal viewing through plane parallel layers, the apparent depth from above is the sum of the apparent thicknesses of each layer:

\text{Apparent depth} = \frac{t_1}{\mu_1} + \frac{t_2}{\mu_2}

where $t_1, t_2$ are real thicknesses of the layers.

So here,

\text{Apparent depth} = \frac{h}{2\sqrt{2}} + \frac{h}{\sqrt{2}}

\frac{h}{2\sqrt{2}} + \frac{h}{\sqrt{2}} = \frac{h}{2\sqrt{2}} + \frac{2h}{2\sqrt{2}} = \frac{3h}{2\sqrt{2}}

Now rationalize/simplify:

\frac{3h}{2\sqrt{2}} = \frac{3h\sqrt{2}}{4}

Thus,

\boxed{\text{Apparent depth} = \frac{3}{4}h\sqrt{2}}

Option C is:

\frac{3}{4}h\sqrt{2}

So the correct option is:

\boxed{\text{C}}

Stored correct answer: C

Our derived answer: C

They match.